Pólya conjecture

The Pólya conjecture is a disproven conjecture in number theory. It involves the Liouville function \(\lambda(n)\), defined as +1 if \(n\) has an even number of prime factors and -1 if \(n\) has an odd number of prime factors, counting multiplicity. The conjecture states that for all \(n > 1\) the summatory Liouville function \(L(n) = \sum_{i = 1}^{n} \lambda(i)\) is always non-positive.

This can be explained visually as follows: Cedric and Royce are standing side by side. At time \(n\), Cedric takes a step forward if \(n\) has an even number of prime factors, and Royce steps forward if \(n\) has an odd number of prime factors. The Pólya conjecture is equivalent to the statement that Royce will always be ahead of Cedric after the starting time.

C. B. Haselgrove disproved the conjecture in 1958 by showing that the sum becomes positive at a number \(n\) estimated at around 1.845 × 10361. Lehman found the first explicit counterexample in 1960, \(L(906180359) = 1\). Tanaka found the smallest counterexample in 1980, \(L(906150257) = 1\). Defining a "crossover" as a point when \(L(n) = 1\) and \(L(n - 1) = 0\), larger crossovers have been found. It is unknown whether there are infinitely many of them.